Online Kernel Dynamic Mode Decomposition for Streaming Time Series Forecasting with Adaptive Windowing

TL;DR

This work tackles real-time forecasting of streaming time series under non-stationarity and strict compute constraints. It introduces WORK-DMD, which combines Random Fourier Feature lifting with online Dynamic Mode Decomposition to model nonlinear dynamics while maintaining a fixed, low per-update cost, facilitated by Sherman–Morrison updates over a rolling window. The approach yields a Koopman-operator-based forecast in a reduced feature space, followed by decoding back to the physical space, and it demonstrates competitive or superior accuracy with a single pass through data across multiple benchmarks, especially for short-horizon predictions. The methods’ sample efficiency and interpretability via eigenstructure support practical deployment on edge devices and in data-constrained environments.

Abstract

Real-time forecasting from streaming data poses critical challenges: handling non-stationary dynamics, operating under strict computational limits, and adapting rapidly without catastrophic forgetting. However, many existing approaches face trade-offs between accuracy, adaptability, and efficiency, particularly when deployed in constrained computing environments. We introduce WORK-DMD (Windowed Online Random Kernel Dynamic Mode Decomposition), a method that combines Random Fourier Features with online Dynamic Mode Decomposition to capture nonlinear dynamics through explicit feature mapping, while preserving fixed computational cost and competitive predictive accuracy across evolving data. WORK-DMD employs Sherman-Morrison updates within rolling windows, enabling continuous adaptation to evolving dynamics from only current data, eliminating the need for lengthy training or large storage requirements for historical data. Experiments on benchmark datasets across several domains show that WORK-DMD achieves higher accuracy than several state-of-the-art online forecasting methods, while requiring only a single pass through the data and demonstrating particularly strong performance in short-term forecasting. Our results show that combining kernel evaluations with adaptive matrix updates achieves strong predictive performance with minimal data requirements. This sample efficiency offers a practical alternative to deep learning for streaming forecasting applications.

Figures (12)

• Figure 1: WORK-DMD methodology pipeline. The method processes multivariate time series data through the following steps: (1) Input multivariate windowed time series $\mathbf{X}_t \in\mathbb{R}^{p\times w}$ with $p$ features, (2) Construction of block-Hankel embedding $\boldsymbol{\mathcal{X}}_t \in \mathbb{R}^{pd \times m}$ to capture temporal correlations across multiple lag orders, (3) Random Fourier Feature (RFF) lifting to transform data into kernel feature space $\boldsymbol{\Psi}_X, \boldsymbol{\Psi}_Y \in \mathbb{R}^{s \times m}$ using Gaussian kernel approximation, (4) Online Dynamic Mode Decomposition update using Sherman-Morrison formulation to maintain Koopman operator $\mathbf{A}_{t+1} \in \mathbb{R}^{s \times s}$ with streaming data by discarding old snapshots (gray) and incorporating new observations (orange), (5) Feature-space forecasting through eigendecomposition in compressed POD basis to generate predictions $\boldsymbol{\Psi}_{\text{pred}} \in \mathbb{C}^{s \times H}$ (red divider separates current features from future predictions), and (6) Decoding via matrix $\mathbf{D}$ to transform feature-space predictions back to physical coordinates $\hat{\mathbf{x}}_{t+h} \in \mathbb{R}^p$. Color coding maintains feature identity throughout the pipeline: original time series features (red, teal, blue) are preserved through Hankel embedding, transformed to kernel features (yellow), processed through online updates, and decoded back to multivariate predictions.
• Figure 2: Time series forecasting comparison on ETTh2 for the Oil Temperature (OT) variable. The figures show 48-hour ahead predictions from WORK-DMD and OneNet against ground truth for two different instances: (a) Instance 1300 and (b) Instance 5700. In both instances, WORK-DMD demonstrates better adherence to the ground truth compared to OneNet.
• Figure 3: Time series forecasting comparison on ETTm1 for the Oil Temperature (OT) variable. The figure shows 1-step ahead predictions comparing WORK-DMD and OneNet against ground truth for instance 500. WORK-DMD exhibits marginally closer alignment to the ground truth, demonstrating detailed tracking of temporal patterns.
• Figure 4: Time series forecasting comparison on WTH dataset for the Wet Bulb Temperature variable. The figures show 24-hour ahead predictions from WORK-DMD and OneNet against ground truth for two different instances: (a) Instance 600 and (b) Instance 5000. In (a), WORK-DMD demonstrates better adherence to the ground truth compared to OneNet, while in (b) both methods exhibit comparable performance. Notably, WORK-DMD showcases adaptive capabilities by accurately forecasting across different temperature regimes, operating in predominantly negative values in (a) and transitioning to positive values in (b).
• Figure 5: Cumulative MSE comparison on ETTh2 dataset. The figures show cumulative mean squared error progression for (a) ETTh2 1-step ahead and (b) ETTh2 48-step ahead forecasting, comparing WORK-DMD and OneNet performance. In (a), WORK-DMD demonstrates faster error correction, resulting in lower cumulative error accumulation compared to OneNet. In (b), both methods exhibit comparable performance across the forecasting horizon.